Piezoelectric beam with magnetic effect and fractional delay term in the internal feedback

This paper analyzes the well-posedness and exponential stability of a fully dynamic piezoelectric beam model incorporating a Caputo fractional derivative with exponential weight as a delay mechanism in the displacement equation. By introducing suitable auxiliary variables, the system is reformulated as an abstract Cauchy problem. The existence and uniqueness of solutions are established via semigroup theory. Using energy estimates and Lyapunov functionals, it is shown that the system is exponentially stable under the condition $\delta_2 \eta^{\sigma - 1} < \delta_1$, linking the delay parameters to the damping coefficients. These results extend previous models by accounting for fractional-order memory effects.